Mandelbrot-Sets-Alternate-Parameter-Planes

Original page : Julia and Mandelbrot Sets. Alternate Parameter Planes by David E. Joyce © 1994.

Images of complex quadratic polynomials under

html verion of this page

z^2+c family

The p-plane is drawn, but initial p parameter is transformed to c:

c = t(p)

So the whole plane (more precisely rectangle viewport) is transformed. For example for c_exponnetial

	plane before transformation 
xMin  = -8.9000000000000004 	 xMax = 0.7000000000000000 
yMin  = -2.3999999999999999 	 yMax = 2.3999999999999999 
	plane after transformation 
xMin  = -1.4010186110735179 	 xMax = 0.6125977074704767 
yMin  = -0.6754631805511510 	 yMax = 0.6754631805511510

z^2 + c

First example is the standard plane, simple: c = p

Description by David E. Joyce

the most notable feature of the set is a cardioid.

z^2 + (k +1/p)

Here

c = k + 1/p

so this transformation consist of 2 transformations:

Translation is performed before inversion. It moves the set with respect to unit circle and thus changes the shape after the whole transformation

See video by Max Million

z^2 + 1/p

Second example is simple inversion about a unit circle centered at origin

Here c = 1/p ( and k = 0 so no translatio)

Description by Nikola Ubavić

composition of the inversion with respect to the unit circle centered at zero, and the conjugation (axial symmetry with respect to the real line). The cardioid from the boundary of the Mandelbrot set in the “standard” parameterization corresponds to the tear-shaped curve in the 1/c parameterization.

Description by David E. Joyce

The inverse of the cardioid is the exterior of a teardrop shape. The circles on the outside of the cardioid are inverted to circles on the inside of the teardrop. The cusp of the cardioid becomes the cusp of the teardrop.

z^2 -2.0+1.0/p

Here : c = -2.0+1.0/p ( and k = -2)

z^2 + 0.25+1.0/p

Here: c = 0.25 + 1.0/p ( and k = 0.25)

Description by Nikola Ubavić

If a 1/4 translation is performed before the inversion, then the cardioid is imaged in a parabola.

The cusp of the cardioid is then moved to the origin, and that inverts to infinity. the cardioid is sent to the outside of a parabola. The circles on the outside of the cardioid are inverted to circles on the inside of the parabola.

Albert Chern:

Since a cardioid is a Möbius inversion of a parabola, you find that the Mandelbrot set is actually a fractal growing along a parabola.

z^2 -1.401155 - 1.0/p

Here: c = -1.401155 - 1.0/p

Exponential map

Here : c = cf + e^p

where

Period doubling cascade:

mz(z-1) = logistic family

mz(1.0-z)

so here m = p

z*(1.0-z)/p

here m = 1/p

(1 + 1/p)z(1.0-z)

here m = 1 + 1/p

Animated gif ( click the image to see the animation)

The code for creating gif file is in src/lcm/inversion_gif/ directory

Compare

Code

Families ( forms) of the complex quadratic polynomial

One (complex) parameter familes of quadratic polynomials. Here parameter space is a 2D plane ( complex plane )

complex double f(const FamilyTypeT FamilyType, const double complex z0 , const complex double p ) {

  	complex double z = z0;
  
  	switch(FamilyType){
	
		case c_type :		{z = z*z + p;  break;}	 // complex quadratic polynomial, p is changed in give_parameter function
		
		case lambda_type: 	{z = p*z*(1.0-z);  break;} // p is changed in give_parameter function
	
	
		default: {z = z*z + p; }
	
	
	}
  
  return  z;
}

Algorithms

Plane transformations

//    transformation = projection = map  from p    to c or lambda parameter
complex double map_parameter(const ProjectionTypeT ProjectionType, const complex double parameter, const complex double translation){

	
	complex double p; 
	// plane transformation 
	switch(ProjectionType){
	
		case identity :{p = translation + parameter;  break;} // first translation and then identity 
		
		case inversion :{p = translation + 1.0/parameter; break;} // first translation then inverion, 2 transformations
		
		case exponentiation :{p = translation + cexp(parameter) ; break;} // here one can change cf to get different image 
		
		
		default: {p = parameter;}
	}
	return p;
}

To do

Log of the Mandelbrot set. Image 8 from Frontiers in complex dynamics by Curtis T. McMullen

Log of the Mandelbrot set. Image 8 from Frontiers in complex dynamics by Curtis T. McMullen

Git

create a new repository on the command line

echo "# Mandelbrot-Sets-Alternate-Parameter-Planes" >> README.md
git init
git add README.md
git commit -m "first commit"
git branch -M main
git remote add origin git@github.com:adammaj1/Mandelbrot-Sets-Alternate-Parameter-Planes.git
git push -u origin main

Local repo

~/Dokumenty/mandelbrot_planes

Subdirectory

mkdir png
git add *.png
git mv  *.png ./png
git commit -m "move"
git push -u origin main

then link the images:

![](./png/n.png "description") 


gitm mv -f

Remove a file/directory from a Git repository without deleting it from the local filesystem

git rm --cached myfile.log

Single directory and all it’s content

git rm --cached -r ./png
git commit -m "png"
git push -u origin main

Rename directory

git add ./lcm/
git mv ./lcm ./png 
git commit -m "png"
git push -u origin main

Github